Supplementary points

Performing a multiway analysis on qEEG data, based on the subjects summary, somehow gives a robust description . A posteriori validation using subject measures can be done using supplementary points technic to compute subjects scores. Supplementary points is similar to ``prediction" as one supposes the description or the model known and want to compute the outcome given the new observation. By analogy of its use in PCA:
let $\sigma_h (s_1 \otimes s_2 \otimes s_3 )$ be a Principal Tensor of the PTA-$3$modes of the data,
$X=P_{S}Y$ [dose*time $\times$lead $\times$band], i.e. $X..(s_1 \otimes s_2 \otimes s_3)=\sigma_h$, one then compute the subject*dose*time's supplementary points of the data $Y$ [subject*dose*time $\times$ lead $\times$ band] by $\tilde{s}_1=\frac{1}{\sigma_h} (Y..(s_2 \otimes s_3))$. If $P_S=P_{S_{dt}} \otimes Id_l \otimes Id_b$ performed the mean over subjects summary, one would have in most of the cases $\tilde{s}_1^{\cdot dt}=1/n_s \sum_s{\tilde{s}_1^{sdt}}=s_1^{dt}$, i.e. one retrieves the component $s_1$. It is then possible to plot standard error of means (SEM) for example, as on the figure fig.8. It is analogue as to do a multivariable regression (subjects*dose*time are the variables) onto one variable $(s_2 \otimes s_3)$. It is mentioned ``in most of the cases" because $X..( s_2 \otimes s_3)$ may not be equal to $\sigma_h s_1$, as the orthogonal decomposition is on the whole space. Then the ``supplementary points" denomination become misleading and will be called pseudo-supplementary points. They will generate a different summary component, sum of the original one plus some orthogonal to it. Looking at the orthogonality constraints in the decomposition it is relatively easy to check that to first Principal Tensor will always generate true supplementary points and every $k$modes Principal Tensor could generate pseudo-components mixing only with previous associated solutions (in the order of $k$modes solutions). Those pseudo-components generated from supplementary points can be also generated from the decomposition itself and may be worthwhile to consider for post-analysis. Another way of performing supplementary points is as follow: let $\sigma_h (s_1 \otimes s_2 \otimes s_3 )$ be a Principal Tensor of the PTA-$3$modes of the data,
$X$ [dose*time $\times$lead $\times$band], i.e. $X..(s_1 \otimes s_2 \otimes s_3)=\sigma_h$, one then compute the subject's supplementary points with the data $Z$ [dose*time $\times$ lead $\times$ band $\times$ subjects] considering $Z=(\sigma_h s_1\otimes s_2 \otimes s_3) \otimes z_4 +\epsilon$ gives $z_4=1/\sigma_h Z..(( s_1\otimes s_2 \otimes s_3)$. Notice that this way every subject ``profile" are proportional, as it is a rank one approximation of the previous method, i.e. $\tilde{s}_1=z_4\otimes s_1 + e$.

Figure 8: Use of supplementary points for SEM plots on the first Principal Tensor (fig.7).